Time Evolution of Holographic Complexity Sotaro Sugishita (Osaka Univ.) based on arxiv:1709.10184 [JHEP 1711, 188 (2017)] with Dean Carmi, Shira Chapman, Hugo Marrochio, Robert Myers RIKEN-Osaka-OIST Joint Workshop 2018 Mar. 12, 2018
AdS/CFT Quantum Gravity on AdS = CFT on boundary of AdS bulk boundary It defines a quantum gravity. How bulk geometry is encoded in the boundary theory.
Geometry and Quantum Information Holographic entanglement entropy Ryu-Takayanagi (2006) minimal codimension 2 surface ~ EE We can study a part of the bulk geometry from EE. Einstein eq. from the properties of EE in CFT
codimension 1 surface (= time slice) Susskind (2014) eternal AdS BH We consider the surface connecting the boundaries = wormhole Like the minimal surface, we extremize the volume. = maximal volume This volume might have the quantum informational meaning!
Time dependence of the volume of the wormhole t
Time dependence of the volume of the wormhole t
Time dependence of the volume of the wormhole t Wormhole grows forever!
How do we see the long-time growth from the dual boundary theory? Can EE capture the feature of the growth?
How do we see the long-time growth from the dual boundary theory? Can EE capture the feature of the growth? No.
EE in eternal AdS black hole Hartman-Maldacena (2013) = CFT 1 CFT 2 Suppose that two CFTs are defined on spaces with the finite volume ~L d 1. Consider EE for two disconnected regions. One is defined on the right boundary and the other is on the left boundary. S EE initially grows in time, but the growth lasts no longer than t~l. However, the wormhole continues growing after t~l.
Entanglement is not enough! Susskind (2014) What is dual to the maximal volume of the wormhole? If such a quantity exists, we can probe the inside of horizon from the boundary theory.
Entanglement is not enough! Susskind (2014) What is dual to the maximal volume of the wormhole? If such a quantity exists, we can probe the inside of horizon from the boundary theory. Complexity?
Complexity Complexity is a quantity characterizing the difficulty to construct a state ψ (or an operator U) from a reference state (or a trivial operator). Roughly, it is a distance between states (or operators). [Iizuka-san will give a talk about complexity.] Recently, there have been many attempts to define complexities for QFTs. 2d CFT [Caputa-Kundu-Miyaji-Takayanagi-Watanabe, 1703.00456] Abelian gauge theories [Hashimoto-Iizuka-SS, 1707.03840] Free scalars [Jefferson-Myers, 1707.08570, Chapman-Heller-Marrochio- Pastawski, 1707.08582],
Maximal volume = complexity? We are not sure if complexity is dual to the maximal volume. Maximal volume itself characterizes the growth of wormhole, and should have an important meaning also in CFT side.
Maximal volume t L t R 1709.10184 Our work: we evaluate the time-dependence of the maximal volume for eternal neutral (and charged) black holes. I don t touch it in this talk.
Complexity = Action? t L t R Wheeler-DeWitt patch Brown, Roberts, Susskind, Swingle, Zhao (2015) dimensionless. more exotic because it touches singularities. Late time behavior is the same as that of the maximal volume. qualitatively the same as the maximal volume? Our work: different time-dependence except for late times.
We evaluate the time-dependence of maximal volume WDW action holographic complexity
Maximal Volume
AdS black hole neutral black holes 1 k = ቐ 0 1 spherical planar hyperbolic L :AdS radius R: curvature radius of boundary or length scale of boundary : position of horizon There are two horizons for small hyperbolic BHs like charged BHs.
Evaluation of the maximal volume 1 From the symmetry, the max. volume is a function of Use the Eddington-Finkelstein coords. To Do: Find v λ, r λ maximizing
Evaluation of the maximal volume 2 does not explicitly depend on is conserved. Use the proper volume parametrization: We can set Turning point :
Evaluation of the maximal volume 3 boundary condition This eq. determines E as a function of the bdry time Then, we can see the time dependence of the max. volume from Finally, we obtain a simple eq. growth rate of volume is finite.
Growth rate of holographic complexity (volume) Holographic complexity Planar black hole (k = 0) blue (d = 4) red (d = 3) green (BTZ, d = 2) monotonically increasing approach to a constant value linear growth of volume at late times, V Mτ
Growth rate of holographic complexity (volume) Spherical black hole (k = 1) d = 4 blue (r h /L = 1) yellow (r h /L = 2) green (r h /L = 5) monotonically increasing approaches to a constant value, but it depends on r h. boundary curvature corrections For large BH, it is the same as planar
Growth rate of holographic complexity (volume) Hyperbolic black hole (k = 1) d = 4 blue (r h /L = 1) yellow (r h /L = 2) green (r h /L = 5) monotonically increasing approaches to a constant value, but it depends on r h. boundary curvature corrections For large BH, it is the same as planar
Wheeler DeWitt action
Wheeler DeWitt patch Evaluate the on-shell action for the Wheeler DeWitt patch. WDW patch: region bounded by null surfaces anchored at the given time-slices on boundaries. null boundaries joints of boundaries (boundary of boundary) We should add the boundary terms and the joint terms. Lehner-Myers-Poisson-Sorkin (2016)
The criterion of the boundary term is that variation principle leads to the Einstein eq. The term is not unique. We have an ambiguity.
Time dependence of WDW action On-shell action depends only on the combination due to the symmetry. We set For planar BHs, the critical times are
Initial times We define the holographic complexity as We use an affine parametrization for null boundaries. WDW action does not depend on the boundary time τ.
Later times WDW action depends on τ. ambiguous parameter To express this only in terms of bdry quantities, we should set : arbitrary bdry length scale
Plots approaches to a constant value not monotonically increasing time derivative = - at blue (r h /L = 1) red (r h /L = 1.5) green (r h /L = 3.5) different from that of max. volume except for late times
Summary We evaluated the time dependence of the maximal volume and WDW action. Similar behaviors at late times, WDW action is ambiguous, although the ambiguity does not change the late time behaviors. Except for late times, very different behaviors. Early time behavior of WDW action is strange (for me). What is the dual? Is it complexity?